Dimensionless Number (or Pure Number)

A dimensionless (or pure) number is a mathematical quantity that has no physical dimensions.

In other words, a pure number is not associated with any unit of measurement, such as meters, seconds, kilograms, or other units in the International System (SI).

Its defining feature is its lack of dimensions, making it especially useful across various fields of physics and mathematics.

It is particularly valuable when expressing relationships between different quantities without the result being influenced by the units of measurement used.

How is a Pure Number Derived?

A pure number is typically derived from the ratio of two quantities that share the same unit of measurement.

In this process, the units cancel out, leaving a result that is free of dimensions.

For example, imagine a rope that is 10 meters long and another that is 2 meters long.

$$ 10 \ m $$

$$ 2 \ m $$

These values are not pure numbers because they are tied to a unit of measurement (meters), meaning they are not dimensionless.

The ratio between the two lengths is:

$$ \frac{10 \, \text{m}}{2 \, \text{m}} = 5 $$

Here, the units (meters) cancel out, leaving just the number 5, which is a pure number.

In other words, the number 5 represents how many times the shorter rope fits into the longer one. It is not associated with any physical dimension.

And so forth.